src/HOL/NanoJava/AxSem.thy

more rules for natural deduction from inequalities

(*  Title:      HOL/NanoJava/AxSem.thy
    Author:     David von Oheimb, Technische Universitaet Muenchen
*)

section "Axiomatic Semantics"

theory AxSem imports State begin

type_synonym assn = "state => bool"
type_synonym vassn = "val => assn"
type_synonym triple = "assn \<times> stmt \<times>  assn"
type_synonym etriple = "assn \<times> expr \<times> vassn"
translations
  (type) "assn" \<leftharpoondown> (type) "state => bool"
  (type) "vassn" \<leftharpoondown> (type) "val => assn"
  (type) "triple" \<leftharpoondown> (type) "assn \<times> stmt \<times> assn"
  (type) "etriple" \<leftharpoondown> (type) "assn \<times> expr \<times> vassn"


subsection "Hoare Logic Rules"

inductive
 hoare :: "[triple set, triple set] => bool"  ("_ |\<turnstile>/ _" [61, 61] 60)
 and ehoare :: "[triple set, etriple] => bool"  ("_ |\<turnstile>\<^sub>e/ _" [61, 61] 60)
 and hoare1 :: "[triple set, assn,stmt,assn] => bool" 
   ("_ \<turnstile>/ ({(1_)}/ (_)/ {(1_)})" [61, 3, 90, 3] 60)
 and ehoare1 :: "[triple set, assn,expr,vassn]=> bool"
   ("_ \<turnstile>\<^sub>e/ ({(1_)}/ (_)/ {(1_)})" [61, 3, 90, 3] 60)
where

  "A  \<turnstile> {P}c{Q} \<equiv> A |\<turnstile> {(P,c,Q)}"
| "A  \<turnstile>\<^sub>e {P}e{Q} \<equiv> A |\<turnstile>\<^sub>e (P,e,Q)"

| Skip:  "A \<turnstile> {P} Skip {P}"

| Comp: "[| A \<turnstile> {P} c1 {Q}; A \<turnstile> {Q} c2 {R} |] ==> A \<turnstile> {P} c1;;c2 {R}"

| Cond: "[| A \<turnstile>\<^sub>e {P} e {Q}; 
            \<forall>v. A \<turnstile> {Q v} (if v \<noteq> Null then c1 else c2) {R} |] ==>
            A \<turnstile> {P} If(e) c1 Else c2 {R}"

| Loop: "A \<turnstile> {\<lambda>s. P s \<and> s<x> \<noteq> Null} c {P} ==>
         A \<turnstile> {P} While(x) c {\<lambda>s. P s \<and> s<x> = Null}"

| LAcc: "A \<turnstile>\<^sub>e {\<lambda>s. P (s<x>) s} LAcc x {P}"

| LAss: "A \<turnstile>\<^sub>e {P} e {\<lambda>v s.  Q (lupd(x\<mapsto>v) s)} ==>
         A \<turnstile>  {P} x:==e {Q}"

| FAcc: "A \<turnstile>\<^sub>e {P} e {\<lambda>v s. \<forall>a. v=Addr a --> Q (get_field s a f) s} ==>
         A \<turnstile>\<^sub>e {P} e..f {Q}"

| FAss: "[| A \<turnstile>\<^sub>e {P} e1 {\<lambda>v s. \<forall>a. v=Addr a --> Q a s};
        \<forall>a. A \<turnstile>\<^sub>e {Q a} e2 {\<lambda>v s. R (upd_obj a f v s)} |] ==>
            A \<turnstile>  {P} e1..f:==e2 {R}"

| NewC: "A \<turnstile>\<^sub>e {\<lambda>s. \<forall>a. new_Addr s = Addr a --> P (Addr a) (new_obj a C s)}
                new C {P}"

| Cast: "A \<turnstile>\<^sub>e {P} e {\<lambda>v s. (case v of Null => True 
                                 | Addr a => obj_class s a \<preceq>C C) --> Q v s} ==>
         A \<turnstile>\<^sub>e {P} Cast C e {Q}"

| Call: "[| A \<turnstile>\<^sub>e {P} e1 {Q}; \<forall>a. A \<turnstile>\<^sub>e {Q a} e2 {R a};
    \<forall>a p ls. A \<turnstile> {\<lambda>s'. \<exists>s. R a p s \<and> ls = s \<and> 
                    s' = lupd(This\<mapsto>a)(lupd(Par\<mapsto>p)(del_locs s))}
                  Meth (C,m) {\<lambda>s. S (s<Res>) (set_locs ls s)} |] ==>
             A \<turnstile>\<^sub>e {P} {C}e1..m(e2) {S}"

| Meth: "\<forall>D. A \<turnstile> {\<lambda>s'. \<exists>s a. s<This> = Addr a \<and> D = obj_class s a \<and> D \<preceq>C C \<and> 
                        P s \<and> s' = init_locs D m s}
                  Impl (D,m) {Q} ==>
             A \<turnstile> {P} Meth (C,m) {Q}"

  \<comment> \<open>\<open>\<Union>Z\<close> instead of \<open>\<forall>Z\<close> in the conclusion and\\
       Z restricted to type state due to limitations of the inductive package\<close>
| Impl: "\<forall>Z::state. A\<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile> 
                            (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm))`Ms ==>
                      A |\<turnstile> (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm))`Ms"

\<comment> \<open>structural rules\<close>

| Asm:  "   a \<in> A ==> A |\<turnstile> {a}"

| ConjI: " \<forall>c \<in> C. A |\<turnstile> {c} ==> A |\<turnstile> C"

| ConjE: "[|A |\<turnstile> C; c \<in> C |] ==> A |\<turnstile> {c}"

  \<comment> \<open>Z restricted to type state due to limitations of the inductive package\<close>
| Conseq:"[| \<forall>Z::state. A \<turnstile> {P' Z} c {Q' Z};
             \<forall>s t. (\<forall>Z. P' Z s --> Q' Z t) --> (P s --> Q t) |] ==>
            A \<turnstile> {P} c {Q }"

  \<comment> \<open>Z restricted to type state due to limitations of the inductive package\<close>
| eConseq:"[| \<forall>Z::state. A \<turnstile>\<^sub>e {P' Z} e {Q' Z};
             \<forall>s v t. (\<forall>Z. P' Z s --> Q' Z v t) --> (P s --> Q v t) |] ==>
            A \<turnstile>\<^sub>e {P} e {Q }"


subsection "Fully polymorphic variants, required for Example only"

axiomatization where
  Conseq:"[| \<forall>Z. A \<turnstile> {P' Z} c {Q' Z};
             \<forall>s t. (\<forall>Z. P' Z s --> Q' Z t) --> (P s --> Q t) |] ==>
                 A \<turnstile> {P} c {Q }"

axiomatization where
  eConseq:"[| \<forall>Z. A \<turnstile>\<^sub>e {P' Z} e {Q' Z};
             \<forall>s v t. (\<forall>Z. P' Z s --> Q' Z v t) --> (P s --> Q v t) |] ==>
                 A \<turnstile>\<^sub>e {P} e {Q }"

axiomatization where
  Impl: "\<forall>Z. A\<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile> 
                          (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm))`Ms ==>
                    A |\<turnstile> (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm))`Ms"

subsection "Derived Rules"

lemma Conseq1: "\<lbrakk>A \<turnstile> {P'} c {Q}; \<forall>s. P s \<longrightarrow> P' s\<rbrakk> \<Longrightarrow> A \<turnstile> {P} c {Q}"
apply (rule hoare_ehoare.Conseq)
apply  (rule allI, assumption)
apply fast
done

lemma Conseq2: "\<lbrakk>A \<turnstile> {P} c {Q'}; \<forall>t. Q' t \<longrightarrow> Q t\<rbrakk> \<Longrightarrow> A \<turnstile> {P} c {Q}"
apply (rule hoare_ehoare.Conseq)
apply  (rule allI, assumption)
apply fast
done

lemma eConseq1: "\<lbrakk>A \<turnstile>\<^sub>e {P'} e {Q}; \<forall>s. P s \<longrightarrow> P' s\<rbrakk> \<Longrightarrow> A \<turnstile>\<^sub>e {P} e {Q}"
apply (rule hoare_ehoare.eConseq)
apply  (rule allI, assumption)
apply fast
done

lemma eConseq2: "\<lbrakk>A \<turnstile>\<^sub>e {P} e {Q'}; \<forall>v t. Q' v t \<longrightarrow> Q v t\<rbrakk> \<Longrightarrow> A \<turnstile>\<^sub>e {P} e {Q}"
apply (rule hoare_ehoare.eConseq)
apply  (rule allI, assumption)
apply fast
done

lemma Weaken: "\<lbrakk>A |\<turnstile> C'; C \<subseteq> C'\<rbrakk> \<Longrightarrow> A |\<turnstile> C"
apply (rule hoare_ehoare.ConjI)
apply clarify
apply (drule hoare_ehoare.ConjE)
apply  fast
apply assumption
done

lemma Thin_lemma: 
  "(A' |\<turnstile>  C         \<longrightarrow> (\<forall>A. A' \<subseteq> A \<longrightarrow> A |\<turnstile>  C       )) \<and> 
   (A'  \<turnstile>\<^sub>e {P} e {Q} \<longrightarrow> (\<forall>A. A' \<subseteq> A \<longrightarrow> A  \<turnstile>\<^sub>e {P} e {Q}))"
apply (rule hoare_ehoare.induct)
apply (tactic "ALLGOALS(EVERY'[clarify_tac \<^context>, REPEAT o smp_tac \<^context> 1])")
apply (blast intro: hoare_ehoare.Skip)
apply (blast intro: hoare_ehoare.Comp)
apply (blast intro: hoare_ehoare.Cond)
apply (blast intro: hoare_ehoare.Loop)
apply (blast intro: hoare_ehoare.LAcc)
apply (blast intro: hoare_ehoare.LAss)
apply (blast intro: hoare_ehoare.FAcc)
apply (blast intro: hoare_ehoare.FAss)
apply (blast intro: hoare_ehoare.NewC)
apply (blast intro: hoare_ehoare.Cast)
apply (erule hoare_ehoare.Call)
apply   (rule, drule spec, erule conjE, tactic "smp_tac \<^context> 1 1", assumption)
apply  blast
apply (blast intro!: hoare_ehoare.Meth)
apply (blast intro!: hoare_ehoare.Impl)
apply (blast intro!: hoare_ehoare.Asm)
apply (blast intro: hoare_ehoare.ConjI)
apply (blast intro: hoare_ehoare.ConjE)
apply (rule hoare_ehoare.Conseq)
apply  (rule, drule spec, erule conjE, tactic "smp_tac \<^context> 1 1", assumption+)
apply (rule hoare_ehoare.eConseq)
apply  (rule, drule spec, erule conjE, tactic "smp_tac \<^context> 1 1", assumption+)
done

lemma cThin: "\<lbrakk>A' |\<turnstile> C; A' \<subseteq> A\<rbrakk> \<Longrightarrow> A |\<turnstile> C"
by (erule (1) conjunct1 [OF Thin_lemma, rule_format])

lemma eThin: "\<lbrakk>A' \<turnstile>\<^sub>e {P} e {Q}; A' \<subseteq> A\<rbrakk> \<Longrightarrow> A \<turnstile>\<^sub>e {P} e {Q}"
by (erule (1) conjunct2 [OF Thin_lemma, rule_format])


lemma Union: "A |\<turnstile> (\<Union>Z. C Z) = (\<forall>Z. A |\<turnstile> C Z)"
by (auto intro: hoare_ehoare.ConjI hoare_ehoare.ConjE)

lemma Impl1': 
   "\<lbrakk>\<forall>Z::state. A\<union> (\<Union>Z. (\<lambda>Cm. (P Z Cm, Impl Cm, Q Z Cm))`Ms) |\<turnstile> 
                 (\<lambda>Cm. (P Z Cm, body Cm, Q Z Cm))`Ms; 
    Cm \<in> Ms\<rbrakk> \<Longrightarrow> 
                A   \<turnstile>  {P Z Cm} Impl Cm {Q Z Cm}"
apply (drule AxSem.Impl)
apply (erule Weaken)
apply (auto del: image_eqI intro: rev_image_eqI)
done

lemmas Impl1 = AxSem.Impl [of _ _ _ "{Cm}", simplified] for Cm

end